It will not be surprising that the steady-state condition is
equivalent to a condition \(\mathcal {L} = 0\)  in the Liouville
representation, a relation suggested by
Eq. (REF ). We show in the following that the
steady-state condition for moments reduces in the large-system limit
to a particular form of the condition \(\mathcal {L} = 0\)  equivalent to
the coherent-state expression in the action (REF ).
However, as is well known in the literature on momentum-space
WKB methods [1]}, [2]}, the
\(\mathcal {L} = 0\)  submanifold contains “momentum” values (our
potential field \(\eta \) ) consistent with derivatives of many different
distributions. To isolate the stationary distribution requires
additional information from boundary conditions; which is the problem
that makes direct solution of higher-dimensional moment equations
difficult. What is gained in passing to the continuum limit is a
non-local relation of the momentum values at each point through
stationary trajectories under variation of the
action (REF ) anchored in the
classical fixed points of the rate equation, a result developed
extensively for the solution of boundary value and first-passage
problems by Freidlin and Wentzell [3]}, and since
generalized to many classes of stochastic
processes [1]}, [2]}.
